Welcome to the CWU Teaching Middle Level Mathematics Community!
This wiki page was created by collaboration of the future middle level mathematics teachers from Central Washington University. The purpose of this page is to create an interactive forum for future teachers to share important focal points in teaching middle level mathematics. Aligned with these focal points are pedagogical ideas and resources.
Guiding Standards for Teaching Middle Level Mathematics
The standards movement is built on the premise that teaching standards have the potential to reform education practices. Teacher-defined standards provide the basis on which teaching establishes expectations for teacher training, professional development, and accountability. Education in the Unitied States looks for guidance from the National Council of Teachers of Mathematics (NCTM) to organize common beliefs, guidelines, and standards among the community of mathematics educators in the Unitied States. NCTM publishing the Curriculum and Evaluation Standards for School Mathematics in 1989 and in 1991 Professional Standards for Teaching Mathematics¹ .
Content Standards
Number and Operations
Teaching Focal Point
Example Questions and Resources
Solve problems involving fractions, decimals, and percents. Rational numbers are in all three notations should be a focal point of middle level mathematics.
Percent question: If Sam got a 7 out of 10, John got a 9 out of 10, and Katie got an 8 out of 10. What was the average PERCENT of the group? Example: Five swimmers are entered into a competition. Four of the swimmers have had their turns. Their scores are 9.8 s, 9.75 s, 9.79 s, and 9.81 s. What score must the last swimmer get in order to win the competition? Both of these websites can be played interactively online or done in the classroom. They have students working with fractions, decimals, and percents interchangeably. Fraction, Decimal, Percent Jeopardy BINGO
Proportional reasoning is a very important part of the middle level mathematics curriculum. In the Common Core State Standards Mathematics it even gains more importance. Modeling is also to be used in every domain so the Modeling with Ratios and Proportions is a great activity to use with your middle level math students.
Recognize and use large numbers, exponential, scientific, and calculator notations. The use of technology and representing large numbers should be taught together.
The same is true of exponential equations, it is best introduce and connected to technology (calculators).
Exponential equations question: Please solve the following equation: 7^3 + 4^2=?
Examples: The websites below are great resources for working with large numbers and exponential equations. Worksheets for exponents Game of Exponents and Polynomials
A game to teaching the order of operation is Order of Operation Game. This game allows teachers flexibliity to emphasize certain aspects of using the rules of order of operation.
A game that can be used to either teach or assess student ability to use exponents numerically is Jeopady Exponent Game. This game is very engaging and can be used in whole group, small groups, or as a individual student assessment activity.
Use and solve problems involving factors, multiples, prime factorization, and prime numbers. These are all number theory concepts and are key to working with rational numbers and later rational concepts connected to algebra.
Factor question: Distribute the factors to solve the equation:
4(3+9)=?
Divisibility and multiple questions:
What is the least number that is divisible by 5, 6 and 15?
Find the LCM of 5, 7 and 9
Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Multiples of 6 are 6, 12, 16, 24, 30
Multiples of 15 are 15, 30, 45, 60
Least Common Multiple (LCM) = 30 Answer: 30
Example: This website is a great resource to introducing these number theory concepts. Factor Tree Interactive Learning
Develop meaning for integers and real numbers.
Represent and compare quantities with integers and real numbers.
When introducing the integers, focus on the idea that negative numbers are additive opposites of these positive pairs.
When introducing the real numbers, focus on the density of numbers. A number line is a good representations for both of these explanations.
Questions for integers: Represent the integers on a number line. How is the integer number line different than the whole number number line?
Questions for rational numbers: Find a rational number between 7/9 and 8/9
Examples: Given a stack of pre-made cards, have students individually draw a card and locate where the integers and real numbers lie a number line drawn on the floor. Students should converse the relationships of the numbers and compare with each other.
For students who are struggling with this concept, the simple game of WAR is a great tool to use. This allows the student to look at the number and compare it to another.
Below are two good resources for teaching integers: Online Integer Games Playing WAR to teach integers
Key numbers properties for middle level mathematics is the associative and commutative properties of addition and multiplication. Also the distributive property of multiplication over addition show be emphasized to simplify computations with integers, fractions, and decimals.
Commutative property question: Balance the equation using the commutative property of addition:
8+29=?
Example: Two great resources for teaching properties. Math Properties Game Grade 3-6 Math Properties Worksheets & Lesson Plans
Question for properties usage: If three dice were rolled, what mathematical properties would you use to arrive at the number 16 if the dice show the numbers ( 3, 4 & 4 )? Number Sense and Operations is an example of how to use the game Tiguous to teach the understanding of operations.
History of Numbers and Operations: Numbers and mathematical operations have always been used. Even in the earliest days of people, some forms of number systems have been in place. Throughout history, the numbers and operations used today have developed immensely and continue to grow. The educational concept of numbers and operations states that students should be able to: understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another; and compute fluently and make reasonable estimates.
Algebra
Teaching Focal Point
Example Questions and Resources
Solve simple equations with variables using algebraic properties. Emphasize the fact the a variable in these equations is only a placeholder, we use the variable to represent a number until we solve for the the number that make the algebraic equation true.
The algebraic properties that should be emphasized are the communitive property of addition and multiplication; the distributive property of multiplication over addition; and the associative property of addition and multiplicaiton.
Students will begin by solving one-step equations.
When this has been mastered, they will move on to solving multistep equations with single variables.
Students should be able to solve equations that are represented by a description, symbols, graph, or table.
Simple equation question: Students want to fill ¼ of a box with apples. The box holds 20 apples. How many apples should the students put in the box? (¼)x = 20 (¼)x ÷ (¼) = 20 ÷ (¼) x = 5 The students should put 5 apples in the box.
Examples and resources:
Students can use manipulatives, such as base 10 strips, to find the missing number in an equation.
Students can play games, such as "Balancing Equations" to learn about algebraic properties.
The attached game is a fun way to teach students of all ages how to recognize symbolic representations of a problem based on the verbal description.
Solve algebraic problems dealing with changing rates such as distance/time and miles per gallon. Represent these values as ratios and percents.
After these skill are introduced in grade 6, students will work to solve word problems
involving rates, ratios, and percentages in grade 7 and master one and two step rate problems in grade 8.
Ratio question: It took a school bus 1 hour and 15 minutes to drive to Ellensburg. Ellensburg is 45 miles away. How fast did the bus travel?
d = 45 miles t = 1.25 hours r = unknown
45 = 1.25r
(45 ÷ 1.25) = (1.25r ÷ 1.25)
36 = r
The bus traveled at 36 mph.
Examples and resources for teaching rate problems:
Students can integrate these concepts with science, and study how precipitation changes in a certain geographical area over a period of time.
Students can find the slope of a ramp, and roll toy cars down the ramp. They can study how the slope affects the time it takes for the car to travel a specific distance.
The following link provides an interactive game in which students use technology to explore and create position vs. time graphs.
Develop a basic understanding of linear functions and how they are represented in an equation and on a graph.
One of the most important step in connecting the equation and graphical representation is to understand how to find the slope of a line in a linear equation and on a graph.
In grade 6, the students are introduced to graphing in the first quadrant, and in grade 7 the students
will graph in all four quadrants of the coordinate plane.
At this point, students will be able to find the slope of any line in equation or graphical representations.
After these skills are introduced and practiced, the students will work to solve word problems
involving one-step and multistep linear equations. Students should be able to find the slope of a line from a table, graph, symbolic descriptions, or verbal description by grade 8.
Finding slope from two points: Find the slope of a line that passes through points (2,1) and (6,3).
y2 = 3 y1 = 1 x2 = 6 x1 = 2
(3 – 1) ÷ (6 – 3) = 2 ÷ 4 = ½
The slope of the line is ½.
Example and resources for teaching linear functions:
Students can integrate math with biology and study the growth pattern of plants. They can graph the growth of their plant, and find the equation for the growth rate.
Students can practice graphing coordinates by playing games such as chess or battleship.
The following link provides a linear graphing activity that utilizes technology. Students will graph equations through an online graphing program, make changes to their linear equations, and analyze how their graphs change.
Functions is a parallel domain in the Common Core State Standards for Mathematics. This is done to remind both teachers and students that functions are a mathematical concept/tool for related mathematical objects in all domains of mathematics. The game Function Machine Game is a great activity for related functions to any mathematical context.
Make connections between multiple representations of the same problem.
For example, students should understand how a table can be represented by an equation or graph.
The four common types of representations students will encounter are verbal descriptions, symbolic descriptions, tables, and graphs. Students are generally taught one or two representations of an equation at a time.
After they understand these representations, more can be added. Eventually students should be able to clearly see the relationship between the representations.
Translate between linear representation questions: Graph the following linear equation y=2x+3
Examples and resources:
The following link provides a wealth of sample problems involving multiple representations.
The following link contains multiple handouts, example problems, definition charts, games, and more. Students will become very experienced in multiple representations of linear relationships.
The History of Algebra:
There are various explanations as to where the word "algebra", which is of Arabic origin, came from. The word is first mentioned in the title of a work by Mahommed be Musa al-Khwarizimi. He lived during the beginning of the 9th century. This contained ideas of comparison and restitution, or comparison and opposition. to reunite and make equal. The full title is ilm al-jebr wa'l-muqabala.
There are other writers that have derived the word from the Arabic part al and gerber,meaning "man". Geber, a Moorish philosopher who was popular in or around the 11th and 12th century, is proposed to be the founder as he perpetuated the name algebra.
The term "algebra" is now in used universally Simply put, it is about finding the unknown. Or you can look it as putting real life problems into mathematical equations and being able to solve them. The goal if algebra is to find out the unknown. Often ending up with x=something. By isolating x on one side of a scale and using = to set both sides of the equation equal, you might have a simple problem such as, Joelee had a basket of apples that she was bringing to her mom, the dog jumped up, knocked the basket out of her hands causing 7 apples to fall on the ground, leaving only 3 in the basket. How many apples did she start with? You would use the following equation to solve this problem: X-7=3 , so in isolating X, it would now read X=7 + 3, meaning that Joelee started with 10 apples.
Algebra develops inductive and deductive reasoning, and the greater your math skills the greater the problems you can solve.
Geometry
Teaching Focal Point
Example Questions
Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes.
It is critical for students to gain knowledge on the properties of each polygon in two- or three-dimensions.
They must learn the necessary terms in order to do so. They need to know the difference between two and three dimensional shapes, identify different shapes, and be able to apply terms (face, vertex, edge, etc.).
They need to know the properties of each shape and be able to understand different classes of shapes.
Shape question: How do an isosceles triangle and a square differ? Compare their properties, such as their internal angles, number of sides, etc. Four-in-a-row is an example of activities that use this game to assess students understanding of important geometry vocabulary.
Examples of Worksheet that can be used as an introduction to geometric shapes and as a diagnostic tool to determine preexisting knowledge of shapes.
Lesson Plan 1 can be used to introduce the concept of coordinates and to tie the new concept of coordinates into the subject matter of geometric shapes and prior knowledge of shapes. This lesson should be used as practice in the area of identifying shapes.
Lesson Plan 2 can be used to introduce surface area and the concept of shapes in two- and three-dimensions.This lesson should be used to show mastery of shape identification and classification and integration of properties in real world situations.
Find the distance between points along horizontal and vertical lines of coordinate system.
As this is closely related to algebra, the knowledge and skills of drawing and reading any geometric shapes is needed for other sub-subjects of mathematics.
Students need to be able to understand how to graph points and read a graph. They need to be able to tell how far apart points are and use the distance formula. They need to be able to graph a line and write an equation for lines.
Calculate distance question: In the given graph above, find out how far point A is from the x- and y- axis.
Lesson Plan 1 can be used as an introduction to and practice of Cartesian coordinates. It can be used to tie coordinate system knowledge to prior knowledge of shapes and lines.
Lesson Plan 2 can be used to practice the concepts of coordinate systems and properties of shapes and lines.
Lesson Plan 3 can be used to show mastery of the concept of line properties and distances on a coordinate system.
Communicating location by the coordinate system is an important idea in middle level mathematics. The game Graphing Battleship is a great activity to use in the classroom to teach and to re-enforce how use the coordinate system to communicate location of objects.
Identify and build a three-dimensional object from two-dimensional representations of that object.
Through this lesson/concept, the students learn to relate how two- and three-dimensional objects are related.
Students need to be able to make three dimensional objects like a cube from a square.
To do this students need to understand the properties of the shapes they are using.
3-D question: Please build a three-dimensional square with each side being 4 inches long.
Lesson Plan 1 can be used to introduce the concept of three-dimensional shapes in relation to two-dimensional shapes. This lesson can be used as a visual introduction to the new concept of dimensions.
Lesson Plan 2 can be used to practice the concepts of three-dimensional shapes in relation to two-dimensional shapes.
Lesson Plan 3 can be used to show mastery of the concept of three-dimensional shapes. It can be used to tie the concept into real world situations, such as astronomy and the solar system.
Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.
It’s important for students to see the relevance of the subject with their life. In addition, they can also practice using the learned materials in everyday life. If students understand how math relates to everyday life, they will respect it more.
Show students how geometry can relate to normal situations, like how shapes can be used to make a painting easier to sketch or how angles can help when playing certain sports.
Application question: How many rectangles are in the American flag?
Lesson Plan 1 can be used as an introduction of polyhedra and geometric shapes in everyday life.
Lesson Plan 2 can be used as a simple example of using math in everyday life. It can be used to tie into the concept of more difficult applications of math.
Lesson Plan 3 can be used to practice properties of geometric shapes and fractals.
Lesson Plan 4 can be used to show mastery of the concepts of geometric relationships.
Example: "Animal Puzzle (Tangram)" game can be used to allow students to utilize and analyze the properties of the given shapes while solving the puzzles (appropriate for all grades since the difficulty level of the puzzles can be changed).
Describe sizes, positions, and orientation of shapes under informal transformations such as flips, turn, slides, and scaling.
They can practice reading the positions of each vertex of a geometric shape, and also recognize the pattern of algebraic computations that can get the results of the transformations.
If a student understands what happens to the position and placement of all parts of a shape when it has been transformed, they will be able to understand they shape and placement better in general.
Students need to know what happens when a shape is transformed and the difference between different transformations.
Transformation question: You are given a triangle where its vertices are located at (1,0), (3,4), and (4, 2). If you flip the triangle 90° clock-wise, and reflect about the y-axis, where are the vertices are now?
Example Worksheet can be used as an introduction to transformations of shapes. They can be used to tie into prior knowledge of shapes and their properties.
Lesson Plan 1 can be used to practice the concept of translations and transformations using technology.
Lesson Plan 2 can be used to practice the concept of translations and transformations using technology. It can be used as a visual representation of the geometric concepts.
Lesson Plan 3 can be used to show mastery of the concept of translations and transformation. It can be used to tie the concepts into real world situations such as city planning and construction.
Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
Similar to learning properties, they can learn to differentiate similar shapes based on unique properties of the shapes. Students need to understand what makes shapes similar.
If they can master how shapes angles, sides, perimeters, areas, or volumes are similar then they will be able to determine subtle differences and really understand the properties of different shapes.
Perimeter question: You have a kite with one side being 5 inches long, and another being 8 inches long. What is its perimeter?
Lesson Plan 1 can be used as an introduction into properties of similar objects. It can be tied into prior knowledge of geometric shapes and objects.
Lesson Plan 2 can be used to practice the concept of area and its relation to objects and shapes.
Lesson Plan 3 can be used to practice the concept of surface area and volume of geometric shapes and objects.
Lesson Plan 4 can be used to show mastery of the concept of similar objects using properties of the objects. It can be used to show the relations to real world situations.
Lesson Plan 5 can be used to show mastery of the concept of similar objects in real world situations such as city construction and planning.
Draw geometric objects with specified properties, such as side lengths or angle measures.
This involves measurements and knowledge of basic knowledge of geometric terms/vocabulary.
Students need to be able to make objects with specific properties.
By following directions students can show how well they understand the objects, by creating them instead of just labeling them.
Geometric properties question: Draw a trapezoid with a pair of 40° angles, another pair of 50° angles, and with a base of 10 inches long.
Example Worksheet can be used as an introduction to geometric properties such as angles.
Lesson Plan 1can be used to practice the concept of angles and lengths in real world situations.
Lesson Plan 2 can be used to show mastery of the concept of geometric properties using fractal properties.
Measurement
Teaching Focal Point
Example Questions
Understand the need for measuring with standard units and become
familiar with standard units in the customary and metric systems.
Students can learn the importance of having standard units (i.e. rather than using one’s arm’s length), and get many opportunities to use different standard units.
Students need to understand why we measure things and how. Students need to know what the different measurements are un the customary and metric systems.
Units of measurement question: If Anna has a 6 inches long ribbon, and Ben has a 10 centimeters long ribbon, how many inches of ribbon do they have combined? Four-in-a-row is a game that can also be used to assess the basic principles of measurement.
Example: Worksheet Example can be used to introduce and practice the concept of measurement and conversion between measurements and measurement systems.
Example:
is a game that can be used for a fun way to have children measure items thrown.
Carry out simple unit conversions with a system of measurements.
Even between standard units, it’s important to use the consistent units to produce an accurate measurement.
Students need to be able to change the unit of measurement between sytems, like from feet to inches.
This is as basic to life as changing hundreds to tens.
Conversion of units of measure question: How long is the line above in feet? There’s 12 inches in 1 foot. How many inches are in 3ft?
Example: Worksheet Example can be used to practice the concept of conversion within a measurement system.
Understand that measurements are approximations and understand how difference in units affects precision.
Students learn that approximations are useful to estimate the outcome of a problem,
but precise measurement gives the most accurate answer to the problem.
They need to understand why we choose different units for measuring different things accurately.
Measure approximation question: Please measure the following line in both inches and centimeters. Compare your answer with your group. Lesson Plan Example can be used to introduce approximation in a measurement system and show the difference in units of measurement.
Solve problems involving scale factors, using ratio and proportion.
Students see the relationship between ratio/proportion to scale factors in measurements.
This also integrates algebra portion of mathematics. Students need to understand what ratio and
proportions are and how they can be used to solve problems.
Scale factor question: Mr. Samson is 6.2ft tall and his arms are 50 inches long. If an artist draws him as a 7 inch-tall person, how long are his arms in the drawing? Lesson Plan Example can be used to show mastery of the concepts of ratio and proportion. It can be used to tie the concepts to real world situations such as Roller Coaster construction.
Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
Students will learn how to choose the most appropriate standard unit to use,
and find different properties of the given geometric shapes (such as area, volume).
If a box has dimension of 3.45in x 4.6in x 10.7cm, what is its volume in cm3? Lesson Plan Example can be used to introduce the concept of volume in relation to area. It can be used as a visual representation of the concept.
The game Formula Fun is a great activity for practicing or assessing student ability to find the surface area or volume of a variety of figures represented pictorially.
Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.
At this level, students know that being precise means to be consistent as well, not just being accurate.
They become more proficient at measuring certain things, and master in finding lengths, areas, volumes, angles, etc.
They need to be able to choose the best tools and formulas to find different properties of shapes.
Measurement question: Please measure the following pyramid using the most appropriate tools: its angles, and lengths of the sides. Lesson Plan Example can be used to show mastery of techniques to find accurate measurements. It can be used to tie the concepts of measurement to real world situations such as city construction and planning.
Solve simple problems involving rates and derived measurements for such attributes as velocity and density.
This focal point utilizes measurement to a more complicated problem,
where students are asked to understand the concept of velocity and density. The basic science (physics) concepts can be integrated.
Velocity question: If Ms. Lee took 30 minutes to travel 15 miles, what was her velocity and how long will she take to travel 120 meters? Worksheet Example can be used to practice simple problems involving rates and measurements. It can be used to tie into real world situations such as walking, running, and driving.
Data Analysis and Probability
Probability
Probability is the calculation of the likelihood of an events occurrence. It is expressed as a number between 0 and 1, 0 being an impossibility and 1 being considered a certainty. Probability can be expressed by the number of occurrences of the targeted event divided by the possible outcomes (number of occurrences plus the number of failures of occurrences). Contemporary applications of probability theory extend human inquiry, and include aspects of computer programming, astrophysics, music, weather prediction, medicine, quantum mechanics, insurance, and study of genetics.
Teaching Focal Point
Example Questions and Resources
Determine probability from a picture.
Connecting the different representations of probability experience is very important.
Probability question: Looking at a picture of a bowl that contains white and black marbles, what is the probability of choosing a
black marble?
Example: This is a website that has an interactive activity. The activity is a spinner with four different sections each a different color. What is the probability of spinning and landing on blue? green? http://www.shodor.org/interactivate/activities/BasicSpinner/
Show Probability Results. Connecting the different representations of probability calculations is very important.
Determine experimental probability of an event. Connecting the experimental and theoretical probabilities is very important.
Probability question: Tim went fishing and caught 12 fish; 8 female and 4 male. What is the experimental probability of a fish caught by Tim being a girl?
Website with a game called, racing game with one die. A game where a dice is rolled and every add number the blue car moves and every even number the red car moves. After one game is played, what is the probability that the same car will win the second game also? http://www.shodor.org/interactivate/activities/RacingGameWithOneDie/
Determine theoretical probability of an event. Students must be able to write and explain how probability is represented mathematically. Students must also be able to make connections between many different representation of the probability of an event.
A crate full of apples contains 20 red apple, 15 green apples, and 8 yellow apples. What is the theoretical probability if a apple is chosen at random it will be red? What is the probability it won't be red?
Lesson plan idea: Give each group of students a bucket containing 20 different colored pieces of paper (4 different colored paper). Have the students find the theoretical probability of choosing each color. Than have the students pick 20 colored pieces of paper at random out of the bucket, each time putting the paper back into the bucket. Have the students discuss their results and how accurate was their theoretical probability?
Represent sample space. Methods of representing the sample space is very important. Some these are a tree diagram, table, or organized list.
Kari has a penny, Don has a nickel, and Bob has a dime. They all flip their coins at the same time. List the possible outcomes using a diagram, list, or chart.
This is a seven day probability interactive project. The students are asked to collect data, interpret data, and show their data through graphs/diagrams/charts ect. http://www2.edc.org/mathpartners/pdfs/6-8%20Statistics%20and%20Probability.pdf
Use theoretical probability to predict experimental
outcomes. This is one of the most important focal points of teaching probability to middle level students. The students must be given many activities where they must make a connection between their mathematical connection and experimental results, so that they can experience the law of large numbers for themselves.
Christian has a bag of 8 yellow marbles, 10 red marbles, and 5 green marbles. After every draw he marks down what color was drawn and puts the marble back in the bag. If Christian does this 100 times, how many times would you expect him to draw a yellow marble?
Website containing lesson plans on finding experimental outcomes using dice, deck of cards, and marbles: http://www.iit.edu/~smile/ma9507.html
Example Game: Elk
Example Game: What's the Prob?
Represent the probability for mutually exclusive events. Student need multiple applications of probability experiences that require the sum of probability events so that students can see the need to identifying mutually exclusive events.
If you have a standard deck of 52 playing cards, what is the probability of drawing a Jack? Queen? A heart? A spade face card?
If you have two fair dice, what is the probability that you will roll snake eyes? an amount equal to 11?
Website containing lesson plans using dice and a deck of cards: http://www.iit.edu/~smile/ma9507.html
History of Probability: The Concept of probability has been around for thousands of years, but probability theory did not arise as a branch of mathematics until the mid-seventeenth century. During the fifteenth century several probability works emerged by Fra Luca Paccioli and Geronimo Cardano. The theory of probability got its start in 1654 when two French Mathematicians, Blaise Pascal and Pierre de Fermat started corresponding through letters about a math problem related to gambling. The gambling problem came up one day when Fermat’s friend who was interested in gambling asked him the question, is there a way that two players who want to finish a game early can divide the winnings fairly based on the chance that each player has of winning the game from that point? Fermat is noted to have completed the first ever rigorous probability calculation when he proved why a gambler has better odds if he bet’s he can roll one six in four rolls rather than betting he can roll double six’s in twenty-four rolls. The Dutch scientist Christian Huygens learned of the correspondence between Pascal and Fermat and in 1657 he published the first book on probability; entitled De Ratiociniis in Ludo Aleae. Because of the appeal of games of chance probability theory became popular. The subject developed rapidly during the 18th century with major contributors Jakob Bernoulli and Abraham de Moivre. In 1812 Pierre de Laplace introduced a host of new ideas and mathematical techniques in his book, Théorie Analytique des Probabilités. Before Laplace, probability theory was solely concerned with developing a mathematical analysis of games of chance. Laplace applied probabilistic ideas to many scientific and practical problems. The 19th century was a time when important applications of probability were developed such as the theory of errors, actuarial mathematics, and statistical mechanics.
Data Analysis
Teaching Focal Point
Example Questions and Resources
Collect data and represent the data in the appropriate graph.
Show the data by the following graphing types: · Bar graphs · Histograms · Stem-and-leaf graphs · Line graphs · Box and whisker
Create a list of each student’s favorite number. Then represent the numbers on a stem- and- leaf graph. By doing this activity students will be able to see the relationship between collecting numbers and making a stem-and-leaf plot.
Ex: The students favorite numbers are: 13,7,18,23,13,8. Next make a stem-and-leaf plot using the collected numbers.
0| 7,8,
1|3,3,8
2|3
Example game activity:
The following activity is an example of a lesson to connect the mode and median to a histogram. It allows students to become more familiar with using mode and median. Histogram activity
Look at a graph and analyze the information. Students should be familiar with mean, median, and mode as part of the analyzing process.
Students should also be able to find the similarities and differences between graphs of related populations.
After reviewing a graph ask the following questions: · What is the mode? · What is the median? · What is the largest number?
An activity on gathering data then analyzing. For gathering data the class will all squeeze each other’s hands and make a graph based on the information. After the graph is made the information is analyzed. Activity
Instead of drilling your students on finding basic statistic use the game MMR to engage students in mastering Mean, Median, and Range.
After reviewing data and a representation of data, base a prediction about the information.
Making predictions based on data and asking questions like “What’s next?” is a good way to test student’s knowledge.
You are given a line graph showing the amount of rain fall over three years. Make a prediction about the amount of rain fall for the fourth year.
This is another example that has a graph of a tub being filled. It’s expressed through a line graph. At the end of the activity the question is asked if it could be a story or another scenario. This allows students to make the predictions off a graph. Tub Graph
Probability and statistics make the most sense when used together to investigate real problems. The investigation Mentos & Soda is a great activity for applying probability and statistics to an engage students 5-12 in true problem solving.
History of Data Analysis
Data Analysis came about in the 17th century. During this time data was collected about death/birth records and crime/mortality. This data would then be analyzed to better improve things like hospital care. Spatial analysis is a category of data analysis and became one of the most frequently used techniques in history. Biologists used it for the study of plants and landscapers started using it for the survey of the vegetation blocks. John snow is most famous for his surveying of cholera.
Calculus
Teaching Focal Point
Example Questions and Resources
Determine the perimeter and area of a composite figure that can be divided into rectangles. (The rectangle formulas need to be emphasized.)
This idea will lead to the area under the curve (integral calculus). Determine the surface area and volume of rectangular prisms using appropriate formulas. Explain why formulas work from a calculus and non-calculus point of view.
Rectangle triangle questions: Use the formula and go to a large number of L and smaller W. A= LW L=length W= width H= height Find the perimeter of a regular polygon and have the number of sides go to infinity. What is the constant multiplied by the length of side?====== = Determine surface area and volume of cylinders using appropriate formulas.
Write an equation that corresponds to a given problem situation and describe a problem situation that corresponds to a given equation.
Max Min question: Given a fence is 1000ft in perimeter. And a farmer wants to divide it in three corrals. What length and width will maximize the area of the corrals? Exponent question idea: Realizing if x is the exponent it will make the constant always increasing. When talking about fractions be sure to include what makes is smaller and what makes it bigger.
Evaluate numerical expressions involving non-negative integer exponents using the laws of exponents. (Understanding when it gets bigger or smaller.) Analyze a problem situation to determine the question(s) to be answered. Apply previously used problem solving strategies in a new context.
Students will be ready to solve problems involving simple negative exponents and should be given the opportunity to do so. Example exponent question: Descriptions of solution processes and explanations can include numbers, words (including mathematical language), pictures, or equations. Students should be able to use all of these representations as needed. Example of applying problem-solving strategies to games: Calculus and Making Connections
History of Calculus:
Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Moscow papyrus (c. 1820 BC), in which an Egyptian mathematician successfully calculated the volume of a pyramidal frustum.
Greek geometers are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create.
Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. It was not until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true.
Indian mathematicians produced a number of works with some ideas of calculus. The formula for the sum of the cubes was first written by Aryabhata circa 500 AD, in order to find the volume of a cube, which was an important step in the development of integral calculus.
Discrete Mathematics
Teaching Focal Point
Example Questions and Resources
Systematic Listing and Counting : Systematic listing and counting is the ability to solve problems such as determining the number of different orders for picking up three friends or counting the number of different computer passwords that are possible with five letters and two numbers such as license plates.
Example question: How many ways can Mark place the six letters R L S T N E, without replacement, in six boxes so that each box contains one letter?
Example activity: In this activity the students will be working on how many different pizza's can be made with eight toppings? http://mathforum.org/workshops/usi/pascal/pizza_pascal.html
This activity should be used to familiarize kids with Pascal's triangle, by way of relating it to what the students enjoy. http://www.lessonplanspage.com/ScienceLAMathMusicCIMDYumaBatClapperRailUnit14 DiscreteMathematicsListingCounting6.htm
This should be used as an introductory lesson to get students understand what it means to systematically count.
Attached is a lesson that uses a game to teach probability and counting. Discrete Mathematics
Vertex-Edge Graphs (Networks)
These graphs provide models for , and lead to solutions of, problems about paths, networks, and relationships among a finite number of elements.
-Find a shortest path from here to there.
-Find a network within a graph that joins all vertices, has no circuits, and has minimum total weight.
-Find a route through a graph that visits each vertex exactly once.
This is a lesson plan on Euler circuits and paths. It specifically deals with Bridges of Konigsberg.
We would use this lesson as a means to reinforce how to use vertex edge graphs. Students will find paths and circuits in this lesson. This will help students grasp the concept of vertex edge graphs as they move into upper division math. This lesson would be best used for sixth through eighth graders. http://www.mathmaniacs.org/lessons/12-euler/index.html
This lesson is about coloring maps, and can be used to introduce vertex edge graphs as well as provide practice and reinforcement. This will be introduced in the fifth grade. The whole purpose of this lesson is to color the map of choice, in as few colors as possible. This should be done following the two rules: 1.) all the states must be colored a different color, and 2.) anybody sharing a border cannot be colored the same color. http://www.mathmaniacs.org/lessons/15-mapcoloring/
Iterative and Recursion
Powerful tools for representing and analyzing regular patterns in sequential step-by-step change, such as day-by-day changes in the chlorine concentration in a swimming pool, year-by-year growth of money in a savings account, or the rising cost of postage as the number of ounces in a package increases.
-How many triangles are present in the 12th stage of Sierpinski’s Triangle.
-Find a formula for the nth stage of Sierpinski’s Triangle.
Students in this activity will be working with iterative and recursive to generalize number patterns and produce a recursion expression.
This activity explores a recurrence relation that has both a geometric and an arithmetic component. This activity could also introduce a discussion of explicit relations. It should be used in grades 6-8 as a method of practice. http://www.colorado.edu/education/DMP/activities/iteration_recursion/lsact05.html
This activity can be used to introduce the concept of recursion as well as to find the nth term of a recursive or iterative pattern or formula. This activity should be used in grades 8-9. http://www.lessonopoly.org/svef/?q=node/2595
History of Discrete Mathematics:
Discrete mathematics is the study of mathematical structures that are more discrete than continuous. It involves iteration and recursion, systematic counting, and graph theory. Sometimes finite math is applied to the field of discrete mathematics especially in the field of business. It is becoming popular in this day and age due to the use of computer sciences. These concepts form computer sciences have helped describe and study objects in computer algorithms and programming languages.
It all started in 1852 with a four color theorem. However, it was not proved until l 1976 when Kenneth Apple and Wolfgang Haken came along. They showed that using computer assistance, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. Also during World War II and the cold war, military used what is known as cryptography, or simply writing in code so that no one person can decode the message. In the 1950’s a telecommunications projected opened that allowed business people to use graph theory and informational theory.
With the use of computer science discrete mathematics is being used more and more each day. This has become an interesting topic for humans in recent decades and is vital that we teach our future students the importance of discrete mathematics.
Process Standards
Problem solving
Teaching Focal Point
Example Questions and Resources
Apply and adapt a variety of strategies to solve problems.
Example question: 1/2 of the students in your class like vanilla ice cream. 1/4 of the students in your class like chocolate ice cream, and the remaining 1/4 like strawberry ice cream. Draw a diagram displaying this information, and label each of the parts appropriately.
Example resource: A Medieval Adventure in Problem-solving
Use a variety of mathematical problem solving skills both in the classroom, and in daily life.
Example question: Johnny goes to the grocery store to buy cereal. When he arrives, he sees that his favorite cereal is 25% off the original price of $4.00. How much is the sale price of his favorite cereal?
Example resource: Problem Solving Mix
Build new mathematical knowledge, through practicing the problem solving skills that have already been mastered.
Example question: Decide whether the following problems are true or false.
1. 2+3 = 3+2
2. 5x4 = 4x5
3. 6+3-2 = 3+2-6
Example resource: An array of problem solving games
Play Tribulation in Number Sense and Operations, a game which challenges students to use three numbers and find four (4) possible answers contained within a word search type formatted puzzle.
Reflect on mathematical problem solving processes.
Example question: Write a journal entry describing the steps you had to take to solve the ice cream problem. What did you have to do first? What did you have to do next? Then give an example of when you would have to use similar problem solving skills to this, in the real world.
Example resource: Click link and then 101 questions for math journals
Reasoning and Proof
Teaching Focal Point
Example Questions and Resources
Recognize and use reasoning and proof as fundamental aspects of doing mathematics. Students should be able to use basic logic to write mathematical arguments.
Measure the sides and angles of two equilateral triangles. Discuss why the angles are the same even though the side lengths are different.
The following link is an excellent resource for teachers who are beginning to teach middle school students about reasoning and proof. It includes sample questions to ask, possible responses, and example problems to use with the students. Beginning to Teach Reasoning and Proof
This link provides sample introductory activities using Geometer's Sketchpad. Students explore how to use proofs as discovery, verification, and systematization. Geometer's Sketchpad Activities
An important part of reasoning and proof is recognizing patterns. This link provides activities for learning how to recognize and build patterns. Patterns are a fundamental piece of reasoning and proof because students learn to build patterns and justify their reasoning. Recognizing Patterns
Make and investigate mathematical conjectures. Students need to frequently tackle a trough questions by making a conjecture, investigating their own conjecture, and then making a conclusion or reflection about what they know to be true.
Write a proof to explain why these two triangles are similar.
In the following link students will work in groups to greate geometric proofs. They will then exchange their proof with another student and try to recreate the proof from the pieces of the original proof. Proof Puzzle
In the following lesson plan students will work with copies of $20 bills to create geometric shapes. Using the proof given studetns will be able to investigate the proof. Hands-On Proof
The following link is for educators. There is background knowledge about conjectures, what they are, how they are used, and their history. Conjecture
Develop and evaluate mathematical arguments and proofs. Student learn how critique a proofs or less formal mathematical argument.
Construct an line that is perpendicular to another line using a compass and straight edge.
This link provides students an opportunity to see how constructions are created. Constructions can be taught as a foundation for teaching proofs. Constructions> >
This link is an introduction to geometry and proofs. There are simple lessons and assessments for students to do. Proof Lesson
In this link, educators will find a large amount of resources for teaching students how to develop and evaluate mathematical arguments. The available activities and student handouts help develop student knowledge of signed number operations, squares, roots, factors, multiples, and primes. Teacher Tips
Select and use various types of mathematical reasoning and methods of proof. Create an appreciation for the many methods of proving a statement logically.
Discuss how an equation, graph, and table can all be used to explain the same problem.
As students move through the curriculum, their method of choosing types of reasoning needs to develop and become more sophisticated. This link provides information about the types of reasoning that should be expanded: algebraic and geometric reasoning, proportional reasoning, probabilistic reasoning, and statistical reasoning. Also provided by this link is information regarding the previously mentioned reasoning and proof objectives. Reasoning and Proof
The following link is a educational resource that teachers can use in order to help them explain and understand reasoning and proofs for their students. nrich proofs
In the middle grades, students sharpen and extend their reasoning skills by analyzing their assertions, using both inductive and deductive reasoning. This link gives examples that teachers can use to help their students build and extend their reasoning skills. Teaching Math
The link below is an example of a unit plan that demonstrates how to use a game to teach students number operations. The powerful way to incorporate this game into a unit on operations is to emphasize the reasoning skills of make and test a conjecture while playing the game.
Students must learn to organize their your own mathematical thinking through many forms of communication.
Example question: There is a family of ten children. Talk amongst yourselves through the process of finding out how many ways you can have 10 children. Explain your process in steps about how you solved this problem. Why do you think that this is the answer? Is your solution the only possible solution?
The two links below provide a lesson plan that gives an example about how we can integrate mathematical thinking into the classroom: http://mathforum.org/brap/wrap2/midlesson.html http://www.readwritethink.org/lessons/lesson_view.asp?id=820
One of the ways to show mathematical understanding is to explain ones solution to a mathematical problem. Students must communicate mathematical thinking with understanding and clearly share with peers, teachers, and others.
Example question: After discovering how to make different ways to come up with 10 children with the choice of a boy and a girl for each child, compare your solution in your small groups. Discuss any differences you got and why? The two links below provide lessons plans, where we can share our mathematical thinking with our peers: http://rachel.ns.purchase.edu/~Jeanine/origami/ http://www.learningpt.org/pdfs/mscLessonPlans/armstrong.pdf Example question: David has been hired to build a house of a good friend. His friend has made the blueprints. If David charges $20 per hour, then how much will his friend have to pay David, knowing that his friend has 40 acres to build on and it will take David and his team and hour and a half to finish building on one acre of land? Work in pairs or groups of three to work out this problem. Communicate with your partner/peers about the strategies to take to complete this problem. Example Game: What's the Prob? Example question: Explain the other players why you made the match you did to the description given? Example game: Four-in-a-row
Not everyone solves a problem the same, so students must be able to compare their own mathematical thinking with a peer or group.
Example question: Have students create their own story problems with systematic counting. When the students have finished their problems, give the problems to another student to work on and try and solve the problem. These students should evaluate the problems and give feedback. The two links below provide a lesson plan example, where we can see how students can create their own mathematical problem and have their classmates try to solve them: http://teachers.net/gazette/FEB09/candler/ http://www.apecneted.org/resources/files/12_5-7_06_9_Wang-Iverson.pdf
Use mathematical language and multiple representations to precisely express ideas of mathematics.
Example question: A wedding couple is planning their wedding reception. They want to use rectangular tables, placing them end-to-end. If one person can sit on either end and two people on each longest side, how many tables do we need to seat 100 people? Write a rule that will determine how many can sit at n tables. Draw a diagram that shows the pattern. Explain how you came to this conclusion for the rule and drawing. The two links below are examples of two lessons plans that provide us with ways that we can use multiple representations and multiple mathematical language to express the ideas of the given problem: http://www.cpm.org/pdfs/information/conference/AC_Con_Mult_Rep.pdf http://www.losmedanos.edu/deved/documents/m25_student_work.multiple_representations_000.pdf
Identify relevant, missing, and extraneous information related to the solution to a problem. Communicate the answer to a problem using appropriate representation and apply a previously used problem-solving strategy in a new context.
Example question: How many ways can you create an equation that will result in the answer 27, using
addition, subtraction, multiplication, division, parentheses, and exponents? Say your answer using correct mathematical language. Example question: Explain, using mathematical language, why you made the match you did to the description given? Example game: Four-in-a-row
Making Connections
Teaching Focal Point
Example Questions and Resources
Recognize and use connections among mathematical ideas. Understand how mathematical ideas interconnect and build on each other to produce the whole idea. Students should see the connection of math concepts and how one idea can be used in a different situation. They should start to see math concepts as a whole, rather than pieces that they just have to get through.
Recognize and apply mathematics in contexts outside of mathematics. Applying mathematics to the world around us gives relevance to mathematics--Modeling should be a part of every mathematics curriculum.
Example questions: The students would roll play that they were doctors in the ER and two patients came into the hospital at the same time. The nurse took blood pressure and heart rates to determine who should be treated first. The students are presented with the information and need to decide if they were the doctor who would they treat first and discuss why? Which gas station should I get gas at? This is a daily concept that the students will be faced with when they can drive. So, say that that I can drive 30 miles to a gas station that gas is 2.70 a gallon or I can drive 60 mile to a gas station that’s gas is 2.40 a gallon. My car gets 60 miles for a gallon. Which gas station would save me the most money. Example resources: Students should connect mathematical concepts to their daily lives, as well as to situations from science, the social sciences, medicine, and commerce. Resource: Connection. 2004. National Council of Teachers of Mathematics. Website: http://standards.nctm.org/document/chapter3/conn.htm The following game involving rate is a natural connection to Calculus and Making Connections
Mathematical Representation
Focal Point
Examples
Create and use representations to organize, record, and communicate mathematical ideas
This is important for middle school teachers because students should know how to solve a problem using a visual representation.
Example question: There are 5 teams in a round-robin tournament. Each team plays each other once. Represent this in a vertex edge graph.
Question 1: Is this a possible path based on if each team plays each other once?
Question 2: Is it possible to not have them cross?
Exploring Geometry lesson plan is a way for students to organize, record and create their own three dimensional figures.
Select, apply, and translate among mathematical representations to solve problems
This is important for middle school teachers because students should know the best way to communicate their ideas in a way that is visual.
Example question: If one is given a diagram, state whether there is a Euler’s circuit and/or path and a Hamilton’s circuit and/or path. If so, give the sequence that represents each one.
Graph Venn Diagrams is helpful in solving mathematical problems when you want to know how many people are taking one subject over another. Here they get to create their own algorithm and solve it on their own.
Pascal’s Triangle is another activity that fits with this focal point. This lesson helps point out Pascal’s uses and what it is to help represent to solve the division and multiplication.
Use representations to model and interpret physical, social, and mathematical phenomena
This is important for middle school teachers because students need to find ways to solve problems using representations.
Example question: There three houses next to each other on a street. Each house needs to be hooked up to the three utilities (water, electricity, and cable), located on a parallel street, one mile away.
Draw a vertex edge graph describing the path from three separate utilities to the three separate homes?
Collecting data and graphing lesson plan is great for kids to work out the problem, see any correlations with the data and then make a graph. Fits perfect with the standard. http://www.theteachersguide.com/lesson%20plans/Math/MPS0001.html Number theory is another lesson plan that is great for students. This helps students develop graphing trees and tables to find probability in a dice game or a card game. This will help with their division and multiplication skills in finding probability. http://www.shodor.org/interactivate/lessons/Combinatorics/
References
¹NCTM, (1991) Professional Standards for Teaching Mathematics, NCTM, Reston, VA
Welcome to the CWU Teaching Middle Level Mathematics Community!
This wiki page was created by collaboration of the future middle level mathematics teachers from Central Washington University. The purpose of this page is to create an interactive forum for future teachers to share important focal points in teaching middle level mathematics. Aligned with these focal points are pedagogical ideas and resources.
Guiding Standards for Teaching Middle Level Mathematics
National Council of Teachers of MathematicsWashington State Endorsement Competencies of Middle Level Mathematics
National Middle School Association Initial Level Teacher Preparation Standards
Purpose of Teaching Standards
The standards movement is built on the premise that teaching standards have the potential to reform education practices. Teacher-defined standards provide the basis on which teaching establishes expectations for teacher training, professional development, and accountability. Education in the Unitied States looks for guidance from the National Council of Teachers of Mathematics (NCTM) to organize common beliefs, guidelines, and standards among the community of mathematics educators in the Unitied States. NCTM publishing the Curriculum and Evaluation Standards for School Mathematics in 1989 and in 1991 Professional Standards for Teaching Mathematics¹ .
Content Standards
Number and Operations
Example: Five swimmers are entered into a competition. Four of the swimmers have had their turns. Their scores are 9.8 s, 9.75 s, 9.79 s, and 9.81 s. What score must the last swimmer get in order to win the competition? Both of these websites can be played interactively online or done in the classroom. They have students working with fractions, decimals, and percents interchangeably.
Fraction, Decimal, Percent Jeopardy BINGO
Proportional reasoning is a very important part of the middle level mathematics curriculum. In the Common Core State Standards Mathematics it even gains more importance. Modeling is also to be used in every domain so the Modeling with Ratios and Proportions is a great activity to use with your middle level math students.
The same is true of exponential equations, it is best introduce and connected to technology (calculators).
Examples: The websites below are great resources for working with large numbers and exponential equations.
Worksheets for exponents Game of Exponents and Polynomials
A game to teaching the order of operation is Order of Operation Game. This game allows teachers flexibliity to emphasize certain aspects of using the rules of order of operation.
A game that can be used to either teach or assess student ability to use exponents numerically is Jeopady Exponent Game. This game is very engaging and can be used in whole group, small groups, or as a individual student assessment activity.
4(3+9)=?
Divisibility and multiple questions:
What is the least number that is divisible by 5, 6 and 15?
Find the LCM of 5, 7 and 9
Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Multiples of 6 are 6, 12, 16, 24, 30
Multiples of 15 are 15, 30, 45, 60
Least Common Multiple (LCM) = 30
Answer: 30
Example: This website is a great resource to introducing these number theory concepts.
Factor Tree Interactive Learning
Represent and compare quantities with integers and real numbers.
When introducing the integers, focus on the idea that negative numbers are additive opposites of these positive pairs.
When introducing the real numbers, focus on the density of numbers. A number line is a good representations for both of these explanations.
Questions for rational numbers: Find a rational number between 7/9 and 8/9
Examples: Given a stack of pre-made cards, have students individually draw a card and locate where the integers and real numbers lie a number line drawn on the floor. Students should converse the relationships of the numbers and compare with each other.
For students who are struggling with this concept, the simple game of WAR is a great tool to use. This allows the student to look at the number and compare it to another.
Below are two good resources for teaching integers:
Online Integer Games Playing WAR to teach integers
8+29=?
Example: Two great resources for teaching properties.
Math Properties Game Grade 3-6
Math Properties Worksheets & Lesson Plans
Question for properties usage: If three dice were rolled, what mathematical properties would you use to arrive at the number 16 if the dice show the numbers ( 3, 4 & 4 )?
Number Sense and Operations is an example of how to use the game Tiguous to teach the understanding of operations.
Algebra
Emphasize the fact the a variable in these equations is only a placeholder,
we use the variable to represent a number until we solve for the the number that make the algebraic equation true.
The algebraic properties that should be emphasized are the communitive property of addition and multiplication; the distributive property of multiplication over addition; and the associative property of addition and multiplicaiton.
Students will begin by solving one-step equations.
When this has been mastered, they will move on to solving multistep equations with single variables.
Students should be able to solve equations that are represented by a description, symbols, graph, or table.
(¼)x = 20
(¼)x ÷ (¼) = 20 ÷ (¼)
x = 5
The students should put 5 apples in the box.
Examples and resources:
- Students can use manipulatives, such as base 10 strips, to find the missing number in an equation.
- Students can play games, such as "Balancing Equations" to learn about algebraic properties.
- The attached game is a fun way to teach students of all ages how to recognize symbolic representations of a problem based on the verbal description.
Equation Game.doc
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- The following link provides an algebraic matching game in which students learn to match solutions with equations.
Example: PEMDAS Puzzle- The link above is a unit plan that addresses solving and creating algebraic equations at the middle school level.
Algebra Matching GameRepresent these values as ratios and percents.
After these skill are introduced in grade 6, students will work to solve word problems
involving rates, ratios, and percentages in grade 7 and master one and two step rate problems in grade 8.
d = 45 miles t = 1.25 hours r = unknown
45 = 1.25r
(45 ÷ 1.25) = (1.25r ÷ 1.25)
36 = r
The bus traveled at 36 mph.
Examples and resources for teaching rate problems:
- Students can integrate these concepts with science, and study how precipitation changes in a certain geographical area over a period of time.
- Students can find the slope of a ramp, and roll toy cars down the ramp. They can study how the slope affects the time it takes for the car to travel a specific distance.
- The following link provides an interactive game in which students use technology to explore and create position vs. time graphs.
Interactive Position v. Time GraphsOne of the most important step in connecting the equation and graphical representation is to understand how to find the slope of a line in a linear equation and on a graph.
In grade 6, the students are introduced to graphing in the first quadrant, and in grade 7 the students
will graph in all four quadrants of the coordinate plane.
At this point, students will be able to find the slope of any line in equation or graphical representations.
After these skills are introduced and practiced, the students will work to solve word problems
involving one-step and multistep linear equations. Students should be able to find the slope of a line from a table, graph, symbolic descriptions, or verbal description by grade 8.
y2 = 3 y1 = 1 x2 = 6 x1 = 2
(3 – 1) ÷ (6 – 3) = 2 ÷ 4 = ½
The slope of the line is ½.
Example and resources for teaching linear functions:
- Students can integrate math with biology and study the growth pattern of plants. They can graph the growth of their plant, and find the equation for the growth rate.
- Students can practice graphing coordinates by playing games such as chess or battleship.
- The following link provides a linear graphing activity that utilizes technology. Students will graph equations through an online graphing program, make changes to their linear equations, and analyze how their graphs change.
Graphing Linear EquationsFunctions is a parallel domain in the Common Core State Standards for Mathematics. This is done to remind both teachers and students that functions are a mathematical concept/tool for related mathematical objects in all domains of mathematics. The game Function Machine Game is a great activity for related functions to any mathematical context.
For example, students should understand how a table can be represented by an equation or graph.
The four common types of representations students will encounter are verbal descriptions, symbolic descriptions, tables, and graphs. Students are generally taught one or two representations of an equation at a time.
After they understand these representations, more can be added. Eventually students should be able to clearly see the relationship between the representations.
Examples and resources:
- The following link provides a wealth of sample problems involving multiple representations.
Multiple Representations Samples- The following link contains multiple handouts, example problems, definition charts, games, and more. Students will become very experienced in multiple representations of linear relationships.
Linear Relations and Multiple Representations- The following link contains a week-long unit in which students learn to relate word problems, tables and line graphs using a graphing calculator.
Multiple Representations Using Graphing CalculatorThere are various explanations as to where the word "algebra", which is of Arabic origin, came from. The word is first mentioned in the title of a work by Mahommed be Musa al-Khwarizimi. He lived during the beginning of the 9th century. This contained ideas of comparison and restitution, or comparison and opposition. to reunite and make equal. The full title is ilm al-jebr wa'l-muqabala.
There are other writers that have derived the word from the Arabic part al and gerber,meaning "man". Geber, a Moorish philosopher who was popular in or around the 11th and 12th century, is proposed to be the founder as he perpetuated the name algebra.
The term "algebra" is now in used universally Simply put, it is about finding the unknown. Or you can look it as putting real life problems into mathematical equations and being able to solve them. The goal if algebra is to find out the unknown. Often ending up with x=something. By isolating x on one side of a scale and using = to set both sides of the equation equal, you might have a simple problem such as, Joelee had a basket of apples that she was bringing to her mom, the dog jumped up, knocked the basket out of her hands causing 7 apples to fall on the ground, leaving only 3 in the basket. How many apples did she start with? You would use the following equation to solve this problem: X-7=3 , so in isolating X, it would now read X=7 + 3, meaning that Joelee started with 10 apples.
Algebra develops inductive and deductive reasoning, and the greater your math skills the greater the problems you can solve.
Geometry
It is critical for students to gain knowledge on the properties of each polygon in two- or three-dimensions.
They must learn the necessary terms in order to do so. They need to know the difference between two and three dimensional shapes, identify different shapes, and be able to apply terms (face, vertex, edge, etc.).
They need to know the properties of each shape and be able to understand different classes of shapes.
Four-in-a-row is an example of activities that use this game to assess students understanding of important geometry vocabulary.
Examples of Worksheet that can be used as an introduction to geometric shapes and as a diagnostic tool to determine preexisting knowledge of shapes.
Lesson Plan 1 can be used to introduce the concept of coordinates and to tie the new concept of coordinates into the subject matter of geometric shapes and prior knowledge of shapes. This lesson should be used as practice in the area of identifying shapes.
Lesson Plan 2 can be used to introduce surface area and the concept of shapes in two- and three-dimensions.This lesson should be used to show mastery of shape identification and classification and integration of properties in real world situations.
As this is closely related to algebra, the knowledge and skills of drawing and reading any geometric shapes is needed for other sub-subjects of mathematics.
Students need to be able to understand how to graph points and read a graph. They need to be able to tell how far apart points are and use the distance formula. They need to be able to graph a line and write an equation for lines.
Lesson Plan 1 can be used as an introduction to and practice of Cartesian coordinates. It can be used to tie coordinate system knowledge to prior knowledge of shapes and lines.
Lesson Plan 2 can be used to practice the concepts of coordinate systems and properties of shapes and lines.
Lesson Plan 3 can be used to show mastery of the concept of line properties and distances on a coordinate system.
Communicating location by the coordinate system is an important idea in middle level mathematics. The game Graphing Battleship is a great activity to use in the classroom to teach and to re-enforce how use the coordinate system to communicate location of objects.
Through this lesson/concept, the students learn to relate how two- and three-dimensional objects are related.
Students need to be able to make three dimensional objects like a cube from a square.
To do this students need to understand the properties of the shapes they are using.
Lesson Plan 1 can be used to introduce the concept of three-dimensional shapes in relation to two-dimensional shapes. This lesson can be used as a visual introduction to the new concept of dimensions.
Lesson Plan 2 can be used to practice the concepts of three-dimensional shapes in relation to two-dimensional shapes.
Lesson Plan 3 can be used to show mastery of the concept of three-dimensional shapes. It can be used to tie the concept into real world situations, such as astronomy and the solar system.
It’s important for students to see the relevance of the subject with their life. In addition, they can also practice using the learned materials in everyday life. If students understand how math relates to everyday life, they will respect it more.
Show students how geometry can relate to normal situations, like how shapes can be used to make a painting easier to sketch or how angles can help when playing certain sports.
Lesson Plan 1 can be used as an introduction of polyhedra and geometric shapes in everyday life.
Lesson Plan 2 can be used as a simple example of using math in everyday life. It can be used to tie into the concept of more difficult applications of math.
Lesson Plan 3 can be used to practice properties of geometric shapes and fractals.
Lesson Plan 4 can be used to show mastery of the concepts of geometric relationships.
Example: "Animal Puzzle (Tangram)" game can be used to allow students to utilize and analyze the properties of the given shapes while solving the puzzles (appropriate for all grades since the difficulty level of the puzzles can be changed).
They can practice reading the positions of each vertex of a geometric shape, and also recognize the pattern of algebraic computations that can get the results of the transformations.
If a student understands what happens to the position and placement of all parts of a shape when it has been transformed, they will be able to understand they shape and placement better in general.
Students need to know what happens when a shape is transformed and the difference between different transformations.
Example Worksheet can be used as an introduction to transformations of shapes. They can be used to tie into prior knowledge of shapes and their properties.
Lesson Plan 1 can be used to practice the concept of translations and transformations using technology.
Lesson Plan 2 can be used to practice the concept of translations and transformations using technology. It can be used as a visual representation of the geometric concepts.
Lesson Plan 3 can be used to show mastery of the concept of translations and transformation. It can be used to tie the concepts into real world situations such as city planning and construction.
Similar to learning properties, they can learn to differentiate similar shapes based on unique properties of the shapes. Students need to understand what makes shapes similar.
If they can master how shapes angles, sides, perimeters, areas, or volumes are similar then they will be able to determine subtle differences and really understand the properties of different shapes.
Lesson Plan 1 can be used as an introduction into properties of similar objects. It can be tied into prior knowledge of geometric shapes and objects.
Lesson Plan 2 can be used to practice the concept of area and its relation to objects and shapes.
Lesson Plan 3 can be used to practice the concept of surface area and volume of geometric shapes and objects.
Lesson Plan 4 can be used to show mastery of the concept of similar objects using properties of the objects. It can be used to show the relations to real world situations.
Lesson Plan 5 can be used to show mastery of the concept of similar objects in real world situations such as city construction and planning.
This involves measurements and knowledge of basic knowledge of geometric terms/vocabulary.
Students need to be able to make objects with specific properties.
By following directions students can show how well they understand the objects, by creating them instead of just labeling them.
Example Worksheet can be used as an introduction to geometric properties such as angles.
Lesson Plan 1can be used to practice the concept of angles and lengths in real world situations.
Lesson Plan 2 can be used to show mastery of the concept of geometric properties using fractal properties.
Measurement
familiar with standard units in the customary and metric systems.
Students can learn the importance of having standard units (i.e. rather than using one’s arm’s length), and get many opportunities to use different standard units.
Students need to understand why we measure things and how. Students need to know what the different measurements are un the customary and metric systems.
Four-in-a-row is a game that can also be used to assess the basic principles of measurement.
Example: Worksheet Example can be used to introduce and practice the concept of measurement and conversion between measurements and measurement systems.
Example:
Even between standard units, it’s important to use the consistent units to produce an accurate measurement.
Students need to be able to change the unit of measurement between sytems, like from feet to inches.
This is as basic to life as changing hundreds to tens.
Example: Worksheet Example can be used to practice the concept of conversion within a measurement system.
Students learn that approximations are useful to estimate the outcome of a problem,
but precise measurement gives the most accurate answer to the problem.
They need to understand why we choose different units for measuring different things accurately.
Lesson Plan Example can be used to introduce approximation in a measurement system and show the difference in units of measurement.
Students see the relationship between ratio/proportion to scale factors in measurements.
This also integrates algebra portion of mathematics. Students need to understand what ratio and
proportions are and how they can be used to solve problems.
Lesson Plan Example can be used to show mastery of the concepts of ratio and proportion. It can be used to tie the concepts to real world situations such as Roller Coaster construction.
Students will learn how to choose the most appropriate standard unit to use,
and find different properties of the given geometric shapes (such as area, volume).
Lesson Plan Example can be used to introduce the concept of volume in relation to area. It can be used as a visual representation of the concept.
The game Formula Fun is a great activity for practicing or assessing student ability to find the surface area or volume of a variety of figures represented pictorially.
At this level, students know that being precise means to be consistent as well, not just being accurate.
They become more proficient at measuring certain things, and master in finding lengths, areas, volumes, angles, etc.
They need to be able to choose the best tools and formulas to find different properties of shapes.
Lesson Plan Example can be used to show mastery of techniques to find accurate measurements. It can be used to tie the concepts of measurement to real world situations such as city construction and planning.
This focal point utilizes measurement to a more complicated problem,
where students are asked to understand the concept of velocity and density. The basic science (physics) concepts can be integrated.
Worksheet Example can be used to practice simple problems involving rates and measurements. It can be used to tie into real world situations such as walking, running, and driving.
Data Analysis and Probability
Probability
Probability is the calculation of the likelihood of an events occurrence. It is expressed as a number between 0 and 1, 0 being an impossibility and 1 being considered a certainty. Probability can be expressed by the number of occurrences of the targeted event divided by the possible outcomes (number of occurrences plus the number of failures of occurrences). Contemporary applications of probability theory extend human inquiry, and include aspects of computer programming, astrophysics, music, weather prediction, medicine, quantum mechanics, insurance, and study of genetics.Connecting the different representations of probability experience is very important.
black marble?
Example: This is a website that has an interactive activity. The activity is a spinner with four different sections each a different color. What is the probability of spinning and landing on blue? green? http://www.shodor.org/interactivate/activities/BasicSpinner/
A website containing a lesson plan on learning the basics of probability:
http://eduref.org/cgi-bin/printlessons.cgi/Virtual/Lessons/Mathematics/Probability/PRB0004.html
Website with a game called, racing game with one die. A game where a dice is rolled and every add number the blue car moves and every even number the red car moves. After one game is played, what is the probability that the same car will win the second game also? http://www.shodor.org/interactivate/activities/RacingGameWithOneDie/
Lesson plan idea: Give each group of students a bucket containing 20 different colored pieces of paper (4 different colored paper). Have the students find the theoretical probability of choosing each color. Than have the students pick 20 colored pieces of paper at random out of the bucket, each time putting the paper back into the bucket. Have the students discuss their results and how accurate was their theoretical probability?
This is a seven day probability interactive project. The students are asked to collect data, interpret data, and show their data through graphs/diagrams/charts ect. http://www2.edc.org/mathpartners/pdfs/6-8%20Statistics%20and%20Probability.pdf
outcomes. This is one of the most important focal points of teaching probability to middle level students. The students must be given many activities where they must make a connection between their mathematical connection and experimental results, so that they can experience the law of large numbers for themselves.
Website containing lesson plans on finding experimental outcomes using dice, deck of cards, and marbles: http://www.iit.edu/~smile/ma9507.html
Example Game: Elk
Example Game: What's the Prob?
If you have two fair dice, what is the probability that you will roll snake eyes? an amount equal to 11?
Website containing lesson plans using dice and a deck of cards: http://www.iit.edu/~smile/ma9507.html
History of Probability:
The Concept of probability has been around for thousands of years, but probability theory did not arise as a branch of mathematics until the mid-seventeenth century. During the fifteenth century several probability works emerged by Fra Luca Paccioli and Geronimo Cardano. The theory of probability got its start in 1654 when two French Mathematicians, Blaise Pascal and Pierre de Fermat started corresponding through letters about a math problem related to gambling. The gambling problem came up one day when Fermat’s friend who was interested in gambling asked him the question, is there a way that two players who want to finish a game early can divide the winnings fairly based on the chance that each player has of winning the game from that point? Fermat is noted to have completed the first ever rigorous probability calculation when he proved why a gambler has better odds if he bet’s he can roll one six in four rolls rather than betting he can roll double six’s in twenty-four rolls. The Dutch scientist Christian Huygens learned of the correspondence between Pascal and Fermat and in 1657 he published the first book on probability; entitled De Ratiociniis in Ludo Aleae. Because of the appeal of games of chance probability theory became popular. The subject developed rapidly during the 18th century with major contributors Jakob Bernoulli and Abraham de Moivre. In 1812 Pierre de Laplace introduced a host of new ideas and mathematical techniques in his book, Théorie Analytique des Probabilités. Before Laplace, probability theory was solely concerned with developing a mathematical analysis of games of chance. Laplace applied probabilistic ideas to many scientific and practical problems. The 19th century was a time when important applications of probability were developed such as the theory of errors, actuarial mathematics, and statistical mechanics.
Data Analysis
Show the data by the following graphing types:
· Bar graphs
· Histograms
· Stem-and-leaf graphs
· Line graphs
· Box and whisker
Ex: The students favorite numbers are: 13,7,18,23,13,8. Next make a stem-and-leaf plot using the collected numbers.
0| 7,8,
1|3,3,8
2|3
Example game activity:
The following activity is an example of a lesson to connect the mode and median to a histogram. It allows students to become more familiar with using mode and median.
Histogram activity
Students should also be able to find the similarities and differences between graphs of related populations.
· What is the mode?
· What is the median?
· What is the largest number?
An activity on gathering data then analyzing. For gathering data the class will all squeeze each other’s hands and make a graph based on the information. After the graph is made the information is analyzed.
Activity
Instead of drilling your students on finding basic statistic use the game MMR to engage students in mastering Mean, Median, and Range.
Making predictions based on data and asking questions like “What’s next?” is a good way to test student’s knowledge.
This is another example that has a graph of a tub being filled. It’s expressed through a line graph. At the end of the activity the question is asked if it could be a story or another scenario. This allows students to make the predictions off a graph.
Tub Graph
Probability and statistics make the most sense when used together to investigate real problems. The investigation Mentos & Soda is a great activity for applying probability and statistics to an engage students 5-12 in true problem solving.
Data Analysis came about in the 17th century. During this time data was collected about death/birth records and crime/mortality. This data would then be analyzed to better improve things like hospital care. Spatial analysis is a category of data analysis and became one of the most frequently used techniques in history. Biologists used it for the study of plants and landscapers started using it for the survey of the vegetation blocks. John snow is most famous for his surveying of cholera.
Calculus
This idea will lead to the area under the curve (integral calculus).
Determine the surface area and volume of rectangular prisms using appropriate formulas.
Explain why formulas work from a calculus and non-calculus point of view.
Use the formula and go to a large number of L and smaller W. A= LW L=length W= width H= height
Find the perimeter of a regular polygon and have the number of sides go to infinity. What is the constant multiplied by the length of side?======
=
Determine surface area and volume of cylinders using appropriate formulas.
Exponent question idea: Realizing if x is the exponent it will make the constant always increasing. When talking about fractions be sure to include what makes is smaller and what makes it bigger.
Analyze a problem situation to determine the question(s) to be answered.
Apply previously used problem solving strategies in a new context.
Example exponent question: Descriptions of solution processes and explanations can include numbers, words (including mathematical language), pictures, or equations. Students should be able to use all of these representations as needed.
Example of applying problem-solving strategies to games:
Calculus and Making Connections
Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Moscow papyrus (c. 1820 BC), in which an Egyptian mathematician successfully calculated the volume of a pyramidal frustum.
Greek geometers are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create.
Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. It was not until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true.
Indian mathematicians produced a number of works with some ideas of calculus. The formula for the sum of the cubes was first written by Aryabhata circa 500 AD, in order to find the volume of a cube, which was an important step in the development of integral calculus.
Discrete Mathematics
Example activity: In this activity the students will be working on how many different pizza's can be made with eight toppings?
http://mathforum.org/workshops/usi/pascal/pizza_pascal.html
This activity should be used to familiarize kids with Pascal's triangle, by way of relating it to what the students enjoy.
http://www.lessonplanspage.com/ScienceLAMathMusicCIMDYumaBatClapperRailUnit14
DiscreteMathematicsListingCounting6.htm
This should be used as an introductory lesson to get students understand what it means to systematically count.
Attached is a lesson that uses a game to teach probability and counting.
Discrete Mathematics
These graphs provide models for , and lead to solutions of, problems about paths, networks, and relationships among a finite number of elements.
-Find a network within a graph that joins all vertices, has no circuits, and has minimum total weight.
-Find a route through a graph that visits each vertex exactly once.
This is a lesson plan on Euler circuits and paths. It specifically deals with Bridges of Konigsberg.
We would use this lesson as a means to reinforce how to use vertex edge graphs. Students will find paths and circuits in this lesson. This will help students grasp the concept of vertex edge graphs as they move into upper division math. This lesson would be best used for sixth through eighth graders.
http://www.mathmaniacs.org/lessons/12-euler/index.html
This lesson is about coloring maps, and can be used to introduce vertex edge graphs as well as provide practice and reinforcement. This will be introduced in the fifth grade. The whole purpose of this lesson is to color the map of choice, in as few colors as possible. This should be done following the two rules: 1.) all the states must be colored a different color, and 2.) anybody sharing a border cannot be colored the same color.
http://www.mathmaniacs.org/lessons/15-mapcoloring/
Powerful tools for representing and analyzing regular patterns in sequential step-by-step change, such as day-by-day changes in the chlorine concentration in a swimming pool, year-by-year growth of money in a savings account, or the rising cost of postage as the number of ounces in a package increases.
-Find a formula for the nth stage of Sierpinski’s Triangle.
Students in this activity will be working with iterative and recursive to generalize number patterns and produce a recursion expression.
This activity explores a recurrence relation that has both a geometric and an arithmetic component. This activity could also introduce a discussion of explicit relations. It should be used in grades 6-8 as a method of practice.
http://www.colorado.edu/education/DMP/activities/iteration_recursion/lsact05.html
This activity can be used to introduce the concept of recursion as well as to find the nth term of a recursive or iterative pattern or formula. This activity should be used in grades 8-9.
http://www.lessonopoly.org/svef/?q=node/2595
History of Discrete Mathematics:
Discrete mathematics is the study of mathematical structures that are more discrete than continuous. It involves iteration and recursion, systematic counting, and graph theory. Sometimes finite math is applied to the field of discrete mathematics especially in the field of business. It is becoming popular in this day and age due to the use of computer sciences. These concepts form computer sciences have helped describe and study objects in computer algorithms and programming languages.
It all started in 1852 with a four color theorem. However, it was not proved until l 1976 when Kenneth Apple and Wolfgang Haken came along. They showed that using computer assistance, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. Also during World War II and the cold war, military used what is known as cryptography, or simply writing in code so that no one person can decode the message. In the 1950’s a telecommunications projected opened that allowed business people to use graph theory and informational theory.
With the use of computer science discrete mathematics is being used more and more each day. This has become an interesting topic for humans in recent decades and is vital that we teach our future students the importance of discrete mathematics.
Process Standards
Problem solving
Example resource:
A Medieval Adventure in Problem-solving
Example resource:
Problem Solving Mix
1. 2+3 = 3+2
2. 5x4 = 4x5
3. 6+3-2 = 3+2-6
Example resource:
An array of problem solving games
Play Tribulation in Number Sense and Operations, a game which challenges students to use three numbers and find four (4) possible answers contained within a word search type formatted puzzle.
Example resource:
Click link and then 101 questions for math journals
Reasoning and Proof
PEMDAS Puzzle
Communicating Mathematically
The two links below provide a lesson plan that gives an example about how we can integrate mathematical thinking into the classroom:
http://mathforum.org/brap/wrap2/midlesson.html
http://www.readwritethink.org/lessons/lesson_view.asp?id=820
The two links below provide lessons plans, where we can share our mathematical thinking with our peers:
http://rachel.ns.purchase.edu/~Jeanine/origami/
http://www.learningpt.org/pdfs/mscLessonPlans/armstrong.pdf
Example question: David has been hired to build a house of a good friend. His friend has made the blueprints. If David charges $20 per hour, then how much will his friend have to pay David, knowing that his friend has 40 acres to build on and it will take David and his team and hour and a half to finish building on one acre of land? Work in pairs or groups of three to work out this problem. Communicate with your partner/peers about the strategies to take to complete this problem.
Example Game: What's the Prob?
Example question: Explain the other players why you made the match you did to the description given?
Example game: Four-in-a-row
The two links below provide a lesson plan example, where we can see how students can create their own mathematical problem and have their classmates try to solve them:
http://teachers.net/gazette/FEB09/candler/
http://www.apecneted.org/resources/files/12_5-7_06_9_Wang-Iverson.pdf
The two links below are examples of two lessons plans that provide us with ways that we can use multiple representations and multiple mathematical language to express the ideas of the given problem:
http://www.cpm.org/pdfs/information/conference/AC_Con_Mult_Rep.pdf
http://www.losmedanos.edu/deved/documents/m25_student_work.multiple_representations_000.pdf
addition, subtraction, multiplication, division, parentheses, and exponents? Say your answer using correct mathematical language.
Example question: Explain, using mathematical language, why you made the match you did to the description given?
Example game: Four-in-a-row
Making Connections
Students should see the connection of math concepts and how one idea can be used in a different situation. They should start to see math concepts as a whole, rather than pieces that they just have to get through.
They could extend their previous knowledge in algebra and data analysis to recognize that the values nearly form a straight line, so C/d is between 3.1 and 3.2 (a rough estimation of Pi).
Lesson examples:
http://teachertipstraining.suite101.com/article.cfm/teaching_mathematics_with_a_box
http://www.teachervision.fen.com/math-principles/lesson-plan/10343.html?detoured=1
Helpful resource site to learn more about how you can link subjects to
math:
http://wiki.nsdl.org/index.php/MiddleSchoolPortal/Connections:_Linking_Mathematics_to_Social_Studies,_Art,_and_Science
The students are presented with the information and need to decide if they were the doctor who would they treat first and discuss why?
Which gas station should I get gas at? This is a daily concept that the students will be faced with when they can drive.
So, say that that I can drive 30 miles to a gas station that gas is 2.70 a gallon or I can drive 60 mile to a gas station that’s gas is 2.40 a gallon. My car gets 60 miles for a gallon. Which gas station would save me the most money.
Example resources: Students should connect mathematical concepts to their daily lives, as well as to situations from science, the social sciences, medicine, and commerce.
Resource: Connection. 2004. National Council of Teachers of Mathematics. Website: http://standards.nctm.org/document/chapter3/conn.htm
The following game involving rate is a natural connection to Calculus and Making Connections
Mathematical Representation
Create and use representations to organize, record, and communicate mathematical ideas
This is important for middle school teachers because students should know how to solve a problem using a visual representation.
Example question: There are 5 teams in a round-robin tournament. Each team plays each other once. Represent this in a vertex edge graph.
Question 1: Is this a possible path based on if each team plays each other once?
Question 2: Is it possible to not have them cross?
Exploring Geometry lesson plan is a way for students to organize, record and create their own three dimensional figures.
http://school.discoveryeducation.com/lessonplans/programs/DM_exploring_geometry/
Population Lab is away where students will analyze some data and create their own representation of the population.
http://www.galeschools.com/lesson_plans/secondary/math/culture.htm
Select, apply, and translate among mathematical representations to solve problems
This is important for middle school teachers because students should know the best way to communicate their ideas in a way that is visual.
Example question: If one is given a diagram, state whether there is a Euler’s circuit and/or path and a Hamilton’s circuit and/or path. If so, give the sequence that represents each one.
Graph Venn Diagrams is helpful in solving mathematical problems when you want to know how many people are taking one subject over another. Here they get to create their own algorithm and solve it on their own.
http://www.shodor.org/interactivate/lessons/AlgorithmDiscovery/
The game Discrete Mathematics uses multiple representations.Pascal’s Triangle is another activity that fits with this focal point. This lesson helps point out Pascal’s uses and what it is to help represent to solve the division and multiplication.
http://www.shodor.org/interactivate/lessons/PascalTriangle/
Use representations to model and interpret physical, social, and mathematical phenomena
This is important for middle school teachers because students need to find ways to solve problems using representations.
Example question: There three houses next to each other on a street. Each house needs to be hooked up to the three utilities (water, electricity, and cable), located on a parallel street, one mile away.
Draw a vertex edge graph describing the path from three separate utilities to the three separate homes?
Collecting data and graphing lesson plan is great for kids to work out the problem, see any correlations with the data and then make a graph. Fits perfect with the standard.http://www.theteachersguide.com/lesson%20plans/Math/MPS0001.html
Number theory is another lesson plan that is great for students. This helps students develop graphing trees and tables to find probability in a dice game or a card game. This will help with their division and multiplication skills in finding probability.
http://www.shodor.org/interactivate/lessons/Combinatorics/
References
¹NCTM, (1991) Professional Standards for Teaching Mathematics, NCTM, Reston, VA